Optimization algorithms for the sound design of chimney and tuning ...

Optimization algorithms for the sound design of chimney and tuning slot organ pipes
Péter Rucz 1 , Fülöp Augusztinovicz 1 , Judit Angster 2 ,
Péter Fiala 1 , Thomas Trommer 2 , András Miklós 3
1 Budapest University of Technology and Economics, H1117 Budapest, Hungary, Email: rucz@hit.bme.hu
2 Fraunhofer-Institut für Bauphysik, 70569 Stuttgart, Germany
3 Steinbeis Transferzentrum Angewandte Akustik, 70499 Stuttgart
Introduction
This contribution presents a novel scaling technique developed
by the authors for the optimal sound design of
chimney organ pipes (see Fig. 1). The goal of the method
is to determine the optimal dimensions of the pipe body
(resonator) for achieving the desired sound spectrum.
Based on the one-dimensional waveguide model of the
resonator the eigenfrequencies of the pipe are calculated
and tuned. The first part of this paper presents the
methodology and the results attained by this technique.
The second part focuses on the development of a similar
optimization algorithm for organ pipes with tuning slot
(Expression). The geometrical irregularity introduced by
the tuning slot requires more advanced modeling techniques
and involves application of numerical methods.
Chimney pipe optimization
The acoustic input admittance function of the pipe is defined
as the ratio of the volume velocity and sound pressure
at the pipe mouth, dependent on the frequency. Its
maxima are located at the eigenfrequencies of the pipe.
The amplitudes of harmonic partials are also affected by
the input admittance function: if a partial is coincident
with a resonance or anti-resonance it is amplified or repressed,
respectively. By choosing the proper dimensions
for the pipe body these effects can be fully exploited and
the sound spectrum of the pipe can be tuned.
Under the cut-on frequency of transversal modes, the resonator
can accurately be described by a one-dimensional
model. This model, as it is shown in Fig. 1, consists of
two waveguides representing the main resonator and the
chimney and the radiation impedances at the open end
and the mouth.
Pipe foot
Z M
Main resonator
Z P0
L P
Z S Z P Z C
Chimney
Z C0
L C
Figure 1: One-dimensional model of a chimney pipe
The input impedance for the complete system Z S can
be written with expressing the input impedance of the
Z R
chimney Z C first:
Z C = Z C0
Z R + iZ C0 tan(kL C )
Z C0 + iZ R tan(kL C )
and (1)
Z S = Z M + Z P0
Z C + iZ P0 tan(kL P )
Z P0 + iZ C tan(kL P ) . (2)
Here L P and L C represent the lengths, while Z P0 and
Z C0 are the acoustic plane wave impedances of the main
resonator and the chimney, respectively. The acoustic
wavenumber is denoted by k. The radiation impedance
at the open end Z R is calculated using the formulas of
Levine and Schwinger [1], while Z M is evaluated using
the relation given by Fletcher and Rossing in [2].
As it is seen, the input admittance function Y S (f) =
1/Z S (f) is dependent on various scaling dimensions. The
set of these parameters is denoted by P.
The optimization objectives are defined as:
1. Tune the fundamental frequency of the pipe f 1 to
a given frequency.
2. Tune an other eigenfrequency to be coincident with
the given harmonic partial, f m = nf 1 , with m and n
denoting the number of the eigenfrequency and the
chosen partial, respectively.
To be able to perform the optimization, a cost function
C(P) is constructed, which measures the distance of the
actual configuration from the ideal one in a special metric.
To minimize C(P) a general global optimization
approach based on the simplex technique developed by
Nelder and Mead [3] is applied. This method shows good
performance and stability for a large set of functions [4].
Various chimney pipes with different optimization objectives
were built using the design process described above.
Original and optimized pipe dimensions are given in Table
1 for one of the experimental pipes. The effect of
the optimization is shown in Fig. 2. As it is seen, the 4 th
eigenfrequency was tuned to the frequency of the 5 th partial,
which was amplified by 15 dB relative to the original
setup. A very good match between the calculated input
admittance and the base line of the steady state sound
spectrum is also observed.
Simulation of the tuning slot
Tuning slots are tuning devices most often used on open
straight metal pipes. By opening the resonator with

Sound pressure measured at mouth [dBSPL]
Table 1: Dimensions of the example chimney pipe
Parameters
Values [mm]
Original Optimized
Chimney length 180.0 56.0
Chimney diameter 19.0 29.3
Resonator length 600.0 852.5
Resonator diameter 79.0
Mouth width 19.0
Mouth height 60.0
110
100
90
80
70
60
50
40
30
20
10
SPL measured at mouth
Calculated admittance
Fundamental: 131 Hz
Fifth partial: 655 Hz
0
0 500 1000 1500
Frequency [Hz]
Figure 2: Comparison of SPL measured at pipe mouth and
optimized input admittance of the example chimney pipe.
a slot, the geometry becomes irregular and the onedimensional
pipe model has to be extended to treat this
discontinuity. Several models were developed and applied
to model the similar case of woodwind instrument
tone holes with success. However, tuning slots differ from
them by their non-circular shape and the thin pipe walls.
Due to these dissimilarities tone hole models can only be
applied for tuning slots as a rough approximation.
Recently, Lefebvre & Scavone [5] proposed a simulation
method to investigate properties of woodwind tone holes
with improved accuracy. Their technique can be interpreted
to tuning slot simulation, as shown in Fig. 3. The
basis of the method is to identify the transfer matrix
(TM) of the tuning slot separate from the pipe by placing
it into a symmetrical cylindrical section, for which
the TM is known. Then the equivalent conentrated parameter
circuit of the slot can be determined and used in
the one-dimensional model.
Validations performed on experimental tuning slot pipes
have shown good correspondence of the simulated input
admittance and measured sound pressure, as displayed in
Fig. 4. These results can serve as the basis of the development
of an optimization method for the sound design
of tuning slot organ pipes, similar to the one presented
for chimney pipes.
Acknowledgments
The financial support of the research by the European
Commission (Grant Agreement Ref# 222104) is grate-
60
40
20
0
−20
−40
Input admittance [dB re. 1/Z P0
]
Sound pressure measured at mouth [dBSPL]
Symmetry plane A
Symmetry plane B
Tuning slot
Domain extension
(PML on surface)
Input plane y
z
x
Figure 3: Arrangement for numerical simulations
110
100
SPL measured at mouth 70
Simulated admittance
90
60
80
50
70
40
60
30
50
20
40
10
30
0
20
−10
10
−20
0
−30
0 500 1000 1500
Frequency [Hz]
Figure 4: Validation of tuning slot simulation by comparison
to measurement
fully acknowledged. P. Rucz acknowledges the support
of the Hungarian research grant TÁMOP - 4.2.2.B-10/1–
2010-0009.
References
[1] H. Levine and J. Schwinger. On the radiation of
sound from an unflanged circular pipe. Physical Review,
73(4):383–406, 1948.
[2] N. H. Fletcher and T. D. Rossing. The physics of
musical instruments, page 475. Springer, 1991.
[3] J. A. Nelder and R. Mead. A simplex method for function
minimalization. The Computer Journal, 7:308–
313, 1965.
[4] J. C. Lagarias, J. A. Reed, M. H. Wright, and P. E.
Wright. Convergence properties of the Nelder-Mead
simplex method in low dimensions. SIAM Journal of
Optimization, 9(1):112–147, 1998.
[5] A. Lefebvre and G. P. Scavone. Refinements to the
model of a single woodwind instrument tonehole. In
Proceedings of 20th International Symposium on Music
Acoustics, Sydney and Katoomba, Australia, August
2010.
Input admittance [dB re. 1/Z P0
]